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## О числе инверсий в исходах схемы перестановок и их анализ с фиксированным числом инверсий

The numerical method of calculation of inversions in permutations uses the method of graphs. As the result we calculate the recurrence for calculation of the numbers of inversions with the use of the bunchs of graphs and finding the probability distribution of the number of inversions with the increes of the size of the permutationThe scheme definded in its name is investigated by asi-ding some outcomes of the similar scheme without restriction. The number of outcomes of the scheme is obtained, the direct enumera-tion is fulfilled, their probability distribution is finded the problem of their numeration is solved and the scheme suggested.

It is proved that any SOо(1, d)-valued cocycle over an ergodic (probability) measurepreserving automorphism is cohomologous to a cocycle having one of three special forms; the recurrence property of such cocycles is also studied.

We define and find a most specific generalization of a fuzzy set of topics assigned to leaves of the rooted tree of a taxonomy. This generalization lifts the set to a “head subject” in the higher ranks of the taxonomy, that is supposed to “tightly” cover the query set, possibly bringing in some errors, both “gaps” and “offshoots”. The method globally minimizes a penalty combining head subjects and gaps and offshoots. We apply this to extract research tendencies from a collection of about 18000 research papers published in Springer journals on data science. We consider a taxonomy of Data Science based on the Association for Computing Machinery Classification of Computing System 2012 (ACM-CCS). We find fuzzy clusters of leaf topics over the text collection and use thematic clusters’ head subjects to make some comments on the tendencies of research.

The aim of this work is to analyze a circle of questions related to the notion of recurrence in general general Markov chains. Being well known in two very different subfields of random systems: lattice random walks and ergodic theory of continuous selfmaps, the recurrence property is next to being neglected in general theory of Markov chains (perhaps except for a few notable exceptions which we will discuss in detail).

The paper proves the recurrence of measurable cocycles with values in the subgroup of SL(2, C) of diagonal and skew-diagonal matrices over an ergodic transformation preserving the probability measure.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

The series Lecture Notes in Computer Science (LNCS), including its subseries Lecture Notes in Artificial Intelligence (LNAI) and Lecture Notes in Bioinformatics (LNBI), has established itself as a medium for the publication of new developments in computer science and information technology research and teaching - quickly, informally, and at a high level.

The two-volume set LNCS 11508 and 11509 constitutes the refereed proceedings of of the 18th International Conference on Artificial Intelligence and Soft Computing, ICAISC 2019, held in Zakopane, Poland, in June 2019.

The 122 revised full papers presented were carefully reviewed and selected from 333 submissions. The papers included in the first volume are organized in the following five parts: neural networks and their applications; fuzzy systems and their applications; evolutionary algorithms and their applications; pattern classification; artificial intelligence in modeling and simulation.

The papers included in the second volume are organized in the following five parts: computer vision, image and speech analysis; bioinformatics, biometrics, and medical applications; data mining; various problems of artificial intelligence; agent systems, robotics and control.

The present research addresses the phenomenon of recurrence at structural, semantic and metatext levels and aims to analyze its impact on the author’s argumentative line and text identity. The paper focuses on the use of prospective and retrospective iteration that plays a particularly significant role in scholarly literature where an academic text is due to comply with specific regulations and to report on the research results in the most objective way. To achieve the goal academic writers rely on employing a system of concepts typical of a particular science, providing unquestionable arguments and critical approach to convince the readers and win their trust. Scholarly articles related to management studies have been exposed to content analysis methodology and the findings reveal that recurrence as a multifunctional phenomenon is an integral part of narrative since it structures the composition, discloses the author’s intention, supplements well justified reasoning through clarification, interpretation, explanation, accentuation and contributes to text consistency, cohesion and coherence.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.